Strange Metal in Magic-Angle Graphene with near Planckian Dissipation

Yuan Cao,1,* Debanjan Chowdhury,1,2,* Daniel Rodan-Legrain,1 Oriol Rubies-Bigorda,1

Kenji Watanabe ,3 Takashi Taniguchi,3 T. Senthil,1,† and Pablo Jarillo-Herrero1,‡1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

2Department of Physics, Cornell University, Ithaca, New York 14853, USA3National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan

(Received 12 November 2019; accepted 23 December 2019; published 18 February 2020)

Recent experiments on magic-angle twisted bilayer graphene have discovered correlated insulatingbehavior and superconductivity at a fractional filling of an isolated narrow band. Here we show that magic-angle bilayer graphene exhibits another hallmark of strongly correlated systems—a broad regime ofT-linear resistivity above a small density-dependent crossover temperature—for a range of fillings near thecorrelated insulator. This behavior is reminiscent of similar behavior in other strongly correlated systems,often denoted “strange metals,” such as cuprates, iron pnictides, ruthenates, and cobaltates, where theobservations are at odds with expectations in a weakly interacting Fermi liquid. We also extract a transport“scattering rate,” which satisfies a near Planckian form that is universally related to the ratio of ðkBT=ℏÞ.Our results establish magic-angle bilayer graphene as a highly tunable platform to investigate strange metalbehavior, which could shed light on this mysterious ubiquitous phase of correlated matter.

DOI: 10.1103/PhysRevLett.124.076801

A panoply of strongly correlated materials have metallicparent states that display properties at odds with theexpectations in a conventional Fermi liquid and are markedby the absence of coherent quasiparticle excitations. Somewell-known families of materials, such as the ruthenates[1,2], cobaltates [3,4], and a subset of the iron-basedsuperconductors [5], show non-Fermi liquid (NFL) behav-ior over a broad intermediate range of temperatures, with acrossover to a conventional Fermi liquid below an emergentlow-energy scale Tcoh, at which coherent electronic qua-siparticles emerge as well-defined excitations. Even morestriking examples of NFL behavior are observed in thehole-doped cuprates [6,7] and certain quantum criticalheavy-fermion compounds [8], where the incoherent fea-tures appear to survive down to the lowest measurabletemperatures, i.e., Tcoh → 0, when superconductivity issuppressed externally. In the incoherent regime, all ofthese materials, in spite of being microscopically distinct,have a resistivity ρðTÞ ∼ T and exhibit a number ofanomalous features [8–11], clearly indicating the absenceof sharp electronic quasiparticles. In strongly interactingnonquasiparticle systems, it has been conjectured [12] thattransport scattering rates (Γ) satisfy a universal “Planckian”bound Γ≲OðkBT=ℏÞ at a temperature T. It is, however,worth noting that in a NFL there is, in general, no cleardefinition for Γ; the scattering rates defined throughdifferent measurements, such as dc and optical conduc-tivity, need not be identical [13]. A microscopic basis forPlanckian limited scattering in generic electronic modelscurrently does not exist; see, however, Refs. [14–16] forsome attempts. One of the most surprising aspects of

incoherent transport in these systems, in spite of thesesubtleties, is that Γ extracted from the dc resistivity(through a procedure specified in Ref. [17]) appears tosatisfy a universal form Γ ¼ CkBT=ℏ with C a number oforder 1 [17,18].Recent experiments [19,20] have reported the discovery

of a correlation driven insulator at fractional fillings (withrespect to a fully filled isolated band) in magic-anglebilayer graphene (MABLG). In MABLG, the relativerotation between two sheets of graphene generates a moirepattern [Fig. 1(a)] with a periodicity that is much larger thanthe underlying interatomic distances in graphene. Thetheoretically estimated electronic bandwidth W is stronglyrenormalized near these small magic angles [21–23]; thenthe strength of the typical Coulomb interactionsU becomesat least comparable to (if not greater than) the bandwidth,U ≳W. Investigating the properties of MABLG as afunction of temperature and carrier density with unprec-edented tunability in a controlled setup can lead to newinsights into the nature of electronic transport in other low-dimensional strongly correlated metallic systems.For our experiments, we have fabricated multiple high-

quality encapsulated MABLG devices [see Fig. 1(a)] usingthe “tear and stack” technique [24,25]. As reported earlier[19,25], we obtain band insulators with a large gap nearn ≈�ns, where n is the carrier density tuned externally byapplying a gate voltage and ns corresponds to four electronsper moire unit cell. On the other hand, correlation driveninsulators with much smaller gap scales [19,20,26] appearnear n ≈�ns=2; we denote these fillings as ν ¼ �2 fromnow on (the fully filled band corresponds to ν ¼ þ4).

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Doping away from these correlated insulators by an addi-tional amount δ, with holes (ν ¼ �2 − δ) and electrons(ν ¼ �2þ δ) leads to superconductivity (SC) [20,26]. Thesuperconducting transition temperature (Tc) measuredrelative to the Fermi energy (εF), as inferred from low-temperature quantum oscillations measurements [20], ishigh, with the largest value of ðTc=εFÞ ∼ 0.07–0.08, indicat-ing strong coupling superconductivity [20]. A schematicν − T phase diagram for MABLG is shown in Fig. 1(b).In this Letter, we investigate the transport phenomenol-

ogy of the metallic states in MABLG as a function ofincreasing temperature over a large range of externallytuned fillings. We have analyzed the temperature depend-ence of the longitudinal dc resistivity ρðTÞ for a number ofdevices (MA1–MA6) over a wide range of fillings, theresults of which appear to be qualitatively similar acrossdevices. In order to highlight the universal aspects of the

behavior, both within and across different samples, weshow the temperature-dependent traces of ρðTÞ for a rangeof different ν in two devices, MA1 and MA4 in Figs. 1(c)and 1(d), respectively. The range of ν chosen for thispurpose is shown as a color bar in Fig. 1(b); see caption fordetails. As it is evident from these traces, at low temper-atures the resistivity exhibits highly nonmonotonic featuresas a function of ν, reflecting the complicated and distinctnature of the ground states [27] [see also inset of Fig. 1(d)].On the other hand, at higher temperatures, the traces forboth devices look qualitatively similar and show a clearmonotonic trend as a function of the gate-tuned filling inspite of the underlying low-temperature differences. As ν istuned externally starting from near the superlattice density,the number of carriers increases, resulting in an enhancedconductivity. This simple picture is to be taken seriouslyonly at high temperatures; at low temperatures the effectivenumber of carriers changes nonmonotonically across ν ¼−2 [20]. We identify the characteristic temperature scaleassociated with the crossover between these distinctregimes, which depends on ν itself, as TcohðνÞ. For a widerange of fillings in the vicinity of ν ¼ �2, we find thatρðTÞ ∼ AT for T ≳ TcohðνÞ, where A denotes the slope ofthe resistivity in units of Ω=K. In a given device, A has arelatively weak dependence on ν, but there is a substantialvariation in the value of A for fillings ν ¼ �2 for a givendevice, as well as in between devices (which have invar-iably slightly different twist angles θ). As we shalldemonstrate below, the linearity of the resistivity is accom-panied by a number of puzzling features, due to which werefer to this regime as a strange metal [28].Figure 2(a) shows ρðTÞ for two separate MABLG

devices, MA3 and MA4, at optimal hole doping, i.e., atfillings where the superconductivity is strongest on thehole-doped side of the correlated insulators at ν ¼ −2 − δ(MA3) and ν ¼ 2 − δ (MA4). From a linear fit to theregime T > Tcoh, we obtain A ≈ 335 Ω=K for MA3 andA ≈ 95 Ω=K for MA4. It is interesting to note that a naiveestimate for the slope of the resistivity, assuming ρðTÞ ∼AT and accounting for the spin and valley degeneracies athigh temperatures, would lead to a value of A ∼ ðh=4e2WÞ,up toOð1Þ prefactors (see Supplemental Material [29]). Foran estimated bandwidth of order W ∼ 10 meV, this givesA ≈ 60 Ω=K, which is not far from the observed values. Wenow compare and contrast the above results for MABLGwith monolayer graphene (MLG), a chemically similar, butweakly correlated, metallic system, which also displaysT-linear resistivity extending over a much wider range oftemperatures. In the inset of Fig. 2(a), we show tworepresentative examples of ρðTÞ for MLG on a SiO2

[33] and hBN [34] substrate, respectively. The resistivityin MLG devices is approximately 2 orders of magnitudesmaller than MABLG, making it a significantly moreconducting metal compared to MABLG. Similarly, theslope of the resistivity for MLG on both substrates is

Correlated insulator

Band insulator B

and

insu

lato

r

Correlated insulatorC

harg

e ne

utra

lity

SC

SC

Filling,0 4-4 -2 2

--2 -2 - 2 2

-3/2 -1/2 3/2

Strange metal

-3 -1 1 3

Strange metal

(a) (b)

(c) (d)

Tem

pera

ture

, T

FIG. 1. Transport in MABLG. (a) Device schematic and four-probe transport measurement configuration (left). (Top right)Side view of the device structure, consisting of bilayers ofgraphene twisted by θ relative to each other, sandwiched byhBN on the top and bottom. The carrier density is tuned using themetal gates at the top and bottom. (Bottom right) SchematicMABLG moire pattern. (b) A schematic phase diagram forMABLG as a function of temperature and filling ν. The strangemetal behavior with T-linear resistivity is primarily restricted tofillings near ν ¼ �2. Orange regions denote superconductivity.Light blue regions denote the correlated insulator region. Thecolor bars indicate the approximate filling ranges investigated andshown in (c) and (d). (c) Resistivity (ρ) as a function oftemperature for device MA1 (θ ¼ 1.16°) for gate-induced den-sities from −1.96 × 1012 (blue) to −1.22 × 1012 cm−2 (red); seehorizontal color bar near ν ¼ −2 in (b). (d) ρðTÞ for device MA4(θ ¼ 1.18°) for gate-induced densities ranging from 1.23 × 1012

(blue) to 1.89 × 1012 cm−2 (red); see horizontal color bar nearν ¼ þ2 in (b). (Inset) Traces for the same device at lowtemperatures. The solid smooth lines have been obtained byusing a Gaussian-weighted filter.

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A ≈ 0.1 Ω=K and is nearly doping independent for a rangeof different densities near charge neutrality. The slope inMLG is thus approximately 3–4 orders of magnitudesmaller than the corresponding slope in MABLG. Whennontwisted bilayer graphene is doped away from chargeneutrality, there is almost no observable temperaturedependence of the resistivity [34,35] and the typical valuesof the resistivity are considerably smaller than in MLG atsimilar densities and temperatures. Thus, a direct compari-son between MABLG and nontwisted BLG sheds light onthe glaring phenomenological differences between thenature of their transport properties. Since there is somevariation in the value of the slopes across different samples,we have studied the dependence of A on the twist angle θ,as shown in Fig. 2(b). The slopes are always evaluated foroptimal doping, i.e., the filling at which Tc is the highest.Orange (cyan) markers denote fillings with ν > 0 (ν < 0),

while solid (empty) symbols denote deviations, δ > 0(δ < 0), away from the correlated insulators at ν ¼ �2.As it is clear from the figure, the slopes at a given θ dependon the sign of ν. Experimentally, it is unclear at whatvalue of θ the magic-angle condition is satisfied precisely.However, we find the highest superconducting Tc aroundθ ∼ 1.07°, which might possibly indicate closest proximityto the magic angle. While we do not have enough datapoints to obtain a clear trend, the typical values of theslopes are large near θ ¼ 1.07°–1.09° (MA5, MA3, respec-tively) and θ ¼ 1.16° (MA1). Fillings ν > 0 also tend tohave higher slopes than ν < 0. We can also obtain theintercept ρ0 ¼ ρðT → 0Þ from an extrapolation of the datafrom high temperatures, which gives us a rough estimate ofthe residual resistivity due to elastic scattering. The resultsare shown in the inset of Fig. 2(b).One of the most interesting aspects of the low-temper-

ature (T < 30 K) longitudinal transport data in MABLG isthat the T-linear resistivity is primarily confined to fillingsnear the correlated insulators at ν ¼ �2� δ with δ≲ 1=2.We note, however, that we do observe a number ofcorrelated phenomena [19,20,26,36,37] and some unusualtransport properties, including T-linear resistivity, also nearν ¼ �1, �3 [marked by asterisks in Fig. 1(b)] in some ofour devices, results that will be reported elsewhere.Figure 2(c) shows the traces of ρðTÞ in the vicinity ofν ¼ −1=2 [see corresponding arrow on the schematic phasediagram in Fig. 1(b)] for device MA4; the traces at ν ¼þ1=2 look qualitatively similar. The inset of Fig. 2(c)shows ρðTÞ near charge neutrality at ν ¼ 0. It is clear thatρðTÞ exhibits qualitatively distinct behavior near thesefillings, without any clear indication of a broad regimeof T-linear resistivity. In Fig. 2(d) and its inset, we plotρðTÞ near ν ¼ −3=2 and ν ¼ þ3=2 for device MA4[fillings marked by arrows in Fig. 1(b)], respectively.While for fillings near ν ¼ −3=2 there is no apparentT-linear resistivity, there is some indication of suchbehavior near ν ¼ 3=2.One of the most celebrated examples of strange metal

behavior is observed in the hole-doped cuprates nearoptimal doping. It shows a number of remarkable features,including ρðTÞ ∼ BT over hundreds of Kelvin without anyapparent signs of crossovers [6,7,17,18], anomalous power-law-dependent optical conductivities [10], or incoherentspectral functions in the presence of a sharp Fermi surface[9]. Focusing on the transport properties, the typical valueof the in-plane sheet resistivity in LSCO (which dependson sample quality) at T ¼ 300 K is approximately ρ ∼0.2–0.3 μΩ cm [6,7], which when normalized by theappropriate interplane distance (d ∼ 6.4 Å) leads to a valueof the resistivity, ρ∼3.1×103–4.7×103Ω. These valuesare roughly comparable to the measured values of theresistivity in MABLG. On the other hand, the value of theslope in La2−xSrxCuO4 (LSCO) is approximately B ∼1.0–2.0 μΩ cm=K [6,7], which when normalized by d leads

(a)

(c)

k

(d)

Twist angle, (º)

(b)

1 1.05 1.1 1.15Twist angle, (º)

0

5

10

0 ()

k

FIG. 2. T-linear resistivity near ν ¼ �2 in MABLG. (a) Resis-tivity (ρ) as a function of temperature for device MA3 (θ ¼ 1.09°)and MA4 (θ ¼ 1.18°) at a gate-tuned density of −1.46 ×1012 cm−2 (ν ¼ −2 − δ) and 1.19 × 1012 cm−2 (ν ¼ 2 − δ), re-spectively. (Inset) ρðTÞ in Ω=K for MLG on SiO2 (green) andhBN (blue), respectively (data from [33,34]). (b) The slopes A ¼dρ=dT obtained at the fillings near ν ¼ �2� δ with optimalsuperconducting Tc for six different devices (MA1–MA6) as afunction of respective twist angles. Orange (cyan) markers denotefillings with ν > 0 (ν < 0), while solid (empty) symbols denotedeviations, δ > 0 (δ < 0), away from the correlated insulators atν ¼ �2. (Inset) Extrapolated resistivity ρ0 for the same devices.(c) ρðTÞ for device MA4 near ν ¼ −1=2 for gate-induceddensities −0.51 × 1012 cm−2 (blue) to −0.29 × 1012 cm−2

(red). The data look similar near ν ¼ þ1=2. (Inset) ρðTÞ forthe same device near ν ¼ 0 and densities between −0.13 ×1012 cm−2 (blue) and 0.13 × 1012 cm−2 (red). (d) ρðTÞ for deviceMA4 near ν ¼ −3=2 for gate-induced densities from −1.28 ×1012 cm−2 (blue) to −1.08 × 1012 cm−2 (red). (Inset) ρðTÞ fordevice MA4 near ν ¼ þ3=2 from 1.08 × 1012 cm−2 (blue) to1.28 × 1012 cm−2 (red).

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to B ∼ 15.6–31.2 Ω=K. The slope is thus slightly smallerthan the typical values in MABLG.Thus far we have compared the typical (nonuniversal)

values of the resistivities and their slopes across differentMABLG devices, as well as with other materials, such asMLG and optimally doped cuprates. We now focus on amore detailed presentation and analysis of device MA2. Webegin by showing the traces of ρðTÞ for MA2 in Fig. 3(a)for a range of ν ¼ −2� δ, roughly corresponding to thecolor bar in Fig. 1(b). As in the other devices, we observethat the resistivity at high temperatures is linear and itdecreases monotonically as ν increases toward chargeneutrality, presumably due to the increase in the numberof carriers measured relative to the superlattice filling. Wehave evaluated the slope A ¼ dρðTÞ=dT at high temper-atures for the same range of ν considered in Fig. 3(a).Figure 3(b) shows the values of A as a function of the gate-induced carrier density for fillings near ν ¼ −2. The slopehere is obtained directly from a linear least-squares fit to theresistivity (see Supplemental Material [29]) for T > 8 K.It is interesting to note that the slope varies nonmonotoni-cally with ν. On the other hand, it is possible that a transportscattering rate (Γ) exhibits more universal character. Inorder to define and obtain Γ from just transport data withinthe same device, we shall now adopt the procedure used inRef. [17]. In the regime of T-linear resistivity, we write thescattering rate, Γ ¼ CkBT=ℏ, and define the numericalprefactor C as a function of ν for device MA2 by assumingthat the resistivity can be well described by a Drude-likeexpression [17,18], such that

C ¼ ℏkB

e2ncð0Þm�ð0Þ A; ð1Þ

where ncð0Þ and m�ð0Þ are the measured density andeffective mass values at low temperatures, T → 0. This

expression gives an operational definition of C and henceof the scattering rate Γ. In device MA2, we have also beenable to extract the actual low-T carrier density [ncð0Þ] andeffective mass [m�ð0Þ] from Shubnikov–de Haas (SdH)quantum oscillations measurements [20,38]. Note thatncð0Þ, as inferred from SdH measurements, is not simplyproportional to ν everywhere in the phase diagram. Instead,for ν ¼ −2 − δ, ncð0Þ corresponds to the gate-induceddensity relative to the correlated insulator at ν ¼ −2[20]. On the other hand, for ν ¼ −2þ δ, ncð0Þ correspondsto the gate-induced density relative to charge neutrality atν ¼ 0 [20], in agreement with low-temperature Hall mea-surements. From the SdH measurements, we can estimatethe Fermi temperature TF ¼ 2πℏ2ncð0Þ=½kBgm�ð0Þ�,where g is the degeneracy factor. It is interesting thatρðTÞ does not exhibit any characteristic changes or cross-overs as the temperature approaches TF for fillings nearν ¼ −2 − δ, where TF ≲ 30 K (see Supplemental Material[29]). In all other examples of strange metals in solid-statesystems, the range over which T-linear resistivity isobserved is significantly below TF. MABLG is thus uniquein that the T-linear resistivity appears to persist to theelectron degeneracy temperature, as inferred from the low-temperature measurements.It is remarkable that C ∼Oð1Þ number for all ν studied

near ν ¼ −2 (see inset of Fig. 3). Moreover, C is weaklydependent on the filling for ν ¼ −2 − δ and varies fromC ∼ 0.2 to 0.4. On the other hand, for ν ¼ −2þ δ, thecoefficient increases monotonically with increasing δ, withC ∼ 1.0–1.6. It is interesting to note that the superconduct-ing Tc is lower for the fillings where Γ is larger (seeSupplemental Material [29]). A similar analysis cannot becarried out asymptotically close to the insulating fillingν ¼ −2 because of the absence of low-temperature SdHmeasurements of m� and nc. However, since the slope inthe metallic regime at high temperatures evolves smoothlyas a function of δ near ν ¼ −2 and the value of nc=m� isalmost independent of the filling, we may naively use theextrapolated value of this ratio for δ → 0 to infer that thehigh-temperature metal at ν ¼ −2 has a similar form ofscattering rate.A similar procedure when used to determine the scatter-

ing rates for other strongly correlated metals [17,18],including ones that lead to unconventional superconduc-tivity at low temperatures, also leads to C ∼Oð1Þ number.A major advantage of our analysis for MABLG is that all ofour measurements are carried out on the same device(MA2) to extract C, as opposed to using data from multipledevices at different dopings. For various members of thecuprate family, the coefficient C ∼ 0.7–1.2, both on thehole- and electron-doped side of the phase diagram [17,18].It is worth mentioning that C ∼Oð1Þ number even formetals like copper, aluminum, and palladium above theirrespective Debye temperatures, where they exhibit aT-linear resistivity due to electron-phonon scattering [17].

(a) (b)

FIG. 3. Planckian scattering in magic-angle bilayer graphene.(a) Resistivity as a function of temperature for device MA2(θ ¼ 1.03°) near ν ¼ −2 for gate-tuned densities from −1.539 ×1012 cm−2 (blue) to −1.113 × 1012 cm−2 (red). The correlatedinsulator at ν ¼ −2 is approximately located near −1.3×1012 cm−2. The solid smooth lines have been obtained by usinga Gaussian-weighted filter. (b) (Left axis) Slope of resistivity as afunction of filling above 8K for the same device evaluated using alinear fit. (Right axis) The coefficient C of the scattering rateΓ ¼ CkBT=ℏ for the same range of fillings.

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In MLG, the T-linear resistivity is also likely due toelectron-phonon scattering above a density-dependent tem-perature [39]. We can calculate the scattering rates for MLGon SiO2 in the regime where they display T-linear resis-tivity. Let us express the resistivity (in units of h=e2) as

ρðTÞ ¼ he2

ℏΓ2εFðVÞ

; ð2Þ

where Γ is as defined earlier and εFðVÞ is controlled by anexternal gate voltage V. This leads to an approximateestimate of C ∼ 0.009–0.022 for the entire range ofdensities studied in Ref. [33], which is a few orders ofmagnitude smaller than the value in MABLG. In Table I,we show a few representative examples of systems wherethe scattering rate has been calculated using the aboveprocedure.We now critically examine some possible mechanisms

that may be responsible for the unusual transport phenom-enology in MABLG. Recently, different electron-phononscattering models have been suggested to try to explain theorigin of T-linear resistivity in twisted BLG near the magicangle [40–42]. These theoretical models are applicableprimarily in the vicinity of fillings ν ¼ 0, for angles close to(but not including) the magic angle, and the coefficient ofthe high-temperature ρ ∼ T regime is then controlled by thestrongly renormalized band structure near charge neutrality(Dirac points). However, in our devices, clear evidence forρ ∼ T and Planckian scattering rates can only be seen nearthe commensurate fillings of ν ¼ �2� δ with δ≲ 1=2 forT < 30 K, but not in the vicinity of charge neutralityjνj ≲ 1=2. Similarly, a recent transport study of twistedBLG devices at angles θ ≳ 2° [43] has reported a power-lawbehavior of the resistivity, ρðTÞ ∼ ρ0 þ Tβ, with a non-monotonic value of β ∼ 0.9–1.7 as a function of filling. Asin our case, superlinear behavior (power-law exponentsubstantially higher than 1) was observed near chargeneutrality. A nonlinear behavior of the resistivity has alsobeen seen very recently near charge neutrality for devices

with angles between 1° and 2° [44]. Moreover, our datashow that TcohðνÞ can be as low as 0.5K [e.g., for deviceMA4 near ν ¼ −2 in Fig. 2(a)], which is an order ofmagnitude lower than the predicted crossover temperatureabove which linear behavior is expected [40]. The maxi-mum value of the slope of the T-linear resistivity, which hasa weak dependence on ν in a given device, is observed nearthe angle (θ ∼ 1.07°) as well as at an angle of θ ∼ 1.16°

[Fig. 2(b)]. However, Refs. [40–42] predict a very strongangle dependence of the slope, with the largest value nearthe magic angle, and this dependence is tied to the strongrenormalization of the Dirac velocity. Although we do notobserve this trend, such a discrepancy could arise as a resultof long-wavelength disorder and slow variations in the twistangle across our devices. In addition, these models seem tounderestimate the slope values and their angular depend-ence substantially, possibly indicating that phonons maynot represent the full contribution to the observed T-linearbehavior. A more sophisticated future theory of electron-phonon scattering that incorporates the following would behighly desirable for a detailed comparison to our exper-imental data: (i) the underlying details of the electronicband structure of MABLG away from charge neutrality,including the strongly nonlinear energy momentumdispersion and presence of van Hove singularities, as wellas (ii) a microscopic treatment of the electron-phonon“gauge” coupling [45,46] and a possible renormalization ofthe deformation potential due to electron-electron inter-actions, and (iii) a folding of the phonon dispersion due tothe moire Brillouin zone, as well as (iv) the possible effectsof density and twist-angle disorder. However, we also notethat, since the T-linear resistivity persists to scales T ∼ TFwithout any signs of a crossover near ν ¼ −2 − δ, while theelectrons are no longer in the degenerate limit, electron-phonon scattering in the metallic state ðT > TcÞ might stillbe insufficient to explain the origin of these unusual results.In principle, T-linear resistivity can arise as a result ofscattering off any electronic collective mode fluctuations(instead of phonons) for TM ≲ T, where TM is a

TABLE I. Materials exhibiting Planckian scattering rates (Γ), where Γ ¼ CkBT=ℏ.

MaterialSlope of resistivity (A) 3D:

μΩ cm=K 2D: Ω=K C Refs.

3D CeCoIn5 1.6 1 [17]CeRu2Si2 0.91 1.1 [17]

ðTMTSFÞ2PF6 (11.8 kbar) 0.38 0.9 [17]UPt3 1.1 1.1 [17]

Cu (T > 100 K) 7 × 10−3 1.0 [17]Au (T > 100 K) 8.4 × 10−3 0.96 [17]

(Quasi-)2D Bi2212 (p ¼ 0.22) 8.0 1.1 [18]LSCO (p ¼ 0.26) 8.2 0.9 [18]PCCO (x ¼ 0.17) 1.7 1.0 [18]MLG on SiO2 0.1 0.01–0.02 [33], This Letter

MABLG 100–300 0.2–1.6 This Letter

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characteristic scale associated with the respective collectivemode, but a crossover across T ∼ TF is neverthelessexpected. It is unclear at the moment if such electroniccollective modes are present near ν ¼ �2.Under conditions of local equilibrium, the conductivity

is given by σ ¼ χDc, where χ ¼ ∂n=∂μ is the electroniccompressibility and Dc ∼ v2=Γ is the charge diffusivity(v≡ a characteristic velocity). Thus far, we have inherentlyassumed that the temperature dependence of the resistivityarises as a result of the temperature dependence ofΓ ∼ kBT=ℏ. An alternative scenario that could, in principle,lead to ρ ∼ T at high temperatures arises solely from thetemperature dependence of the compressibility, χ ∼ 1=T, ina system with a bounded kinetic energy. However, in someof our devices, the ρ ∼ AT behavior persists from hightemperatures down to temperatures as low as 0.5 K withoutany change in the slope A [e.g., device MA4 in Fig. 2(a)].Moreover, as emphasized earlier, ρ does not show anycrossovers as the temperature is swept through the esti-mated εF for fillings near ν ¼ −2 − δ. Thus, while it isunlikely that the above mechanism is responsible for theobserved temperature dependence of the resistivity, asystematic future study of the charge compressibility inMABLG as a function of increasing temperature is highlydesirable. Instead of relying on the dc transport measure-ments to extract Γ, it will be interesting to measure andanalyze the extent to which the optical scattering rates inMABLG satisfy a universal form in future experiments.Interestingly, recent measurements of diffusivity and com-pressibility in a cold-atoms-based experiment [47] findρ ∼ T from temperatures higher than the bandwidth downto lower temperatures without any signs of a crossover.In light of these results, any successful theory has to

account for the following universal aspects of the phenom-enology: (i) a T-linear resistivity with values Oðh=e2Þ inthe vicinity of commensurate fillings, ν ¼ �2, with nearPlanckian [C ∼Oð1Þ] scattering rates, (ii) a weak depend-ence of the slope of the resistivity on ν, (iii) relativeinsensitivity of the resistivity to TF and, by extension, to theunderlying details of the Fermi-surface at low temperatures,and (iv) the presence of a small Tcoh above which thetransport is unconventional. Since Tcoh can be as low as 0.5Kin some of our devices, it will be interesting to see if futureexperiments find evidence of T-linear resistivity down toeven lower temperatures (i.e., Tcoh → 0). If so, it is possiblethat the NFL behavior in MABLG is controlled by a T ¼ 0quantum critical point or quantum critical phase; there is notenough conclusive experimental information available atpresent to elaborate on this aspect. On the other hand, if Tcohis nonzero, it is likely that themetallic regime ofMABLG forfillings near ν ¼ �2 realizes an intermediate-scale NFL.Assuming that the temperature remains smaller than thetypical interaction strengths but large compared to Tcoh, thestate is neither a classical liquid (or gas) nor a degeneratequantum liquid. The system can then be best described as a

“semiquantum” liquid, with no coherent quasiparticle exci-tations and no sharply defined Fermi surface. Describingsuch a regime in a theoretically controlled limit is challeng-ing, but recent progress has been made in the study of somemodels [13,48], which find evidence of such incoherentbehavior. Guided by these studies and by the presentexperiments, we will further develop microscopic theoriesof transport in such a regime elsewhere.

We acknowledge Helena Briones for the assistance onthe figures for the device structure. This work has beenprimarily supported by the National Science Foundation(DMR-1809802), the Center for Integrated QuantumMaterials under NSF Grant No. DMR-1231319, and theGordon and Betty Moore Foundation’s EPiQS Initiativethrough Grant No. GBMF4541 to P. J.-H. for devicefabrication, transport measurements, and data analysis.This work was performed in part at the HarvardUniversity Center for Nanoscale Systems (CNS), a memberof the National Nanotechnology Coordinated InfrastructureNetwork (NNCI), which is supported by the NationalScience Foundation under NSF ECCS GrantNo. 1541959. D. C. received support from a Gordon andBetty Moore Foundation Fellowship, under the EPiQSinitiative, Grant No. GBMF4303, at MIT. D. R.-L acknowl-edges support from Fundació Bancaria “la Caixa” (LCF/BQ/AN15/10380011) and from the U.S. Army ResearchOffice Grant No. W911NF-17-S-0001. O. R.-B acknowl-edges support from Fundació Privada Cellex. T. S. issupported by a U.S. Department of Energy AwardNo. DE-SC0008739, and in part by a SimonsInvestigator award from the Simons Foundation. K.W.and T. T. acknowledge support from the Elemental StrategyInitiative conducted by the MEXT, Japan, A3 Foresight byJSPS and the CREST (JPMJCR15F3), JST.

*These authors contributed equally.†senthil@mit.edu‡pjarillo@mit.edu

[1] N. E. Hussey, A. P. Mackenzie, J. R. Cooper, Y. Maeno, S.Nishizaki, and T. Fujita, Phys. Rev. B 57, 5505 (1998).

[2] L. Klein, J. S. Dodge, C. H. Ahn, G. J. Snyder, T. H.Geballe, M. R. Beasley, and A. Kapitulnik, Phys. Rev. Lett.77, 2774 (1996).

[3] S. Y. Li, L. Taillefer, D. G. Hawthorn, M. A. Tanatar, J.Paglione, M. Sutherland, R. W. Hill, C. H. Wang, and X. H.Chen, Phys. Rev. Lett. 93, 056401 (2004).

[4] Y. Wang, N. S. Rogado, R. J. Cava, and N. P. Ong, Nature(London) 423, 425 (2003).

[5] T. Shibauchi, A. Carrington, and Y. Matsuda, Annu. Rev.Condens. Matter Phys. 5, 113 (2014).

[6] H. Takagi, B. Batlogg, H. L. Kao, J. Kwo, R. J. Cava,J. J. Krajewski, and W. F. Peck, Phys. Rev. Lett. 69, 2975(1992).

[7] P. Giraldo-Gallo, J. A. Galvis, Z. Stegen, K. A. Modic, F. F.Balakirev, J. B. Betts, X. Lian, C. Moir, S. C. Riggs, J. Wu,

PHYSICAL REVIEW LETTERS 124, 076801 (2020)

076801-6

A. T. Bollinger, X. He, I. Božović, B. J. Ramshaw, R. D.McDonald, G. S. Boebinger, and A. Shekhter, Science 361,479 (2018).

[8] H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev.Mod. Phys. 79, 1015 (2007).

[9] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys.75, 473 (2003).

[10] D. v. d. Marel, H. J. A. Molegraaf, J. Zaanen, Z. Nussinov, F.Carbone, A. Damascelli, H. Eisaki, M. Greven, P. H. Kes,and M. Li, Nature (London) 425, 271 (2003).

[11] S.-C. Wang, H.-B. Yang, A. K. P. Sekharan, H. Ding, J. R.Engelbrecht, X. Dai, Z. Wang, A. Kaminski, T. Valla, T.Kidd, A. V. Fedorov, and P. D. Johnson, Phys. Rev. Lett. 92,137002 (2004).

[12] J. Zaanen, Nature (London) 430, 512 (2004).[13] D. Chowdhury, Y. Werman, E. Berg, and T. Senthil, Phys.

Rev. X 8, 031024 (2018).[14] M. Blake, Phys. Rev. Lett. 117, 091601 (2016).[15] T. Hartman, S. A. Hartnoll, and R. Mahajan, Phys. Rev. Lett.

119, 141601 (2017).[16] A. A. Patel and S. Sachdev, Phys. Rev. Lett. 123, 066601

(2019).[17] J. A. N. Bruin, H. Sakai, R. S. Perry, and A. P. Mackenzie,

Science 339, 804 (2013).[18] A. Legros, S. Benhabib, W. Tabis, F. Laliberte, M. Dion, M.

Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Z. Li, P.Auban-Senzier, N. Doiron-Leyraud, P. Fournier, D. Colson,L. Taillefer, and C. Proust, Nat. Phys. 15, 142 (2019).

[19] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y.Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi,E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Nature(London) 556, 80 (2018).

[20] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E.Kaxiras, and P. Jarillo-Herrero, Nature (London) 556, 43(2018).

[21] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. CastroNeto, Phys. Rev. Lett. 99, 256802 (2007).

[22] E. Suárez Morell, J. D. Correa, P. Vargas, M. Pacheco, andZ. Barticevic, Phys. Rev. B 82, 121407(R) (2010).

[23] R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci.U.S.A. 108, 12233 (2011).

[24] K. Kim, A. DaSilva, S. Huang, B. Fallahazad, S. Larentis, T.Taniguchi, K. Watanabe, B. J. LeRoy, A. H. MacDonald, andE. Tutuc, Proc. Natl. Acad. Sci. U.S.A. 114, 3364 (2017).

[25] Y. Cao, J. Y. Luo, V. Fatemi, S. Fang, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, and P.Jarillo-Herrero, Phys. Rev. Lett. 117, 116804 (2016).

[26] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K.Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R.Dean, Science 363, 1059 (2019).

[27] It is interesting to note that, for both devices at fillings nearν ¼ −2 − δ (MA1) and ν ¼ 2þ δ (MA4), we observe abroad shoulder around 5 K where the resistivity drops by asubstantial fraction. The suppression in the resistivity,before the eventual onset of SC, could be a signature ofstrong local pairing fluctuations. We will elaborate on thiselsewhere.

[28] The term strange metal is typically used to denote a widevariety of unusual metallic behavior, seemingly at odds withthe expectations in a weakly interacting Fermi liquid. All

strange metals do not necessarily have identical reasons forthe strangeness. Our usage of the term is in keeping with thispractice.

[29] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.124.076801 for detailsof sample preparation and data analysis, which includesRefs. [30–32].

[30] J. Orenstein, G. A. Thomas, A. J. Millis, S. L. Cooper, D. H.Rapkine, T. Timusk, L. F. Schneemeyer, and J. V. Waszczak,Phys. Rev. B 42, 6342 (1990).

[31] C. C. Homes, S. V. Dordevic, M. Strongin, D. A. Bonn, R.Liang, W. N. Hardy, S. Komiya, Y. Ando, G. Yu, N. Kaneko,X. Zhao, M. Greven, D. N. Basov, and T. Timusk, Nature(London) 430, 539 (2004).

[32] T. Hazra, N. Verma, and M. Randeria, Phys. Rev. X 9,031049 (2019).

[33] J.-H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer,Nat. Nanotechnol. 3, 206 (2008).

[34] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang,S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim,K. L. Shepard, and J. Hone, Nat. Nanotechnol. 5, 722(2010).

[35] S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F.Schedin, D. C. Elias, J. A. Jaszczak, and A. K. Geim, Phys.Rev. Lett. 100, 016602 (2008).

[36] A. L. Sharpe, E. J. Fox, A.W. Barnard, J. Finney, K.Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Science 365, 605 (2019).

[37] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das,C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A.Bachtold, A. H. MacDonald, and D. K. Efetov, Nature(London) 574, 653 (2019).

[38] In principle, the effective mass that enters transport quan-tities can be different from the effective cyclotron mass, asdeduced from quantum oscillations measurements.

[39] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev.Mod. Phys. 83, 407 (2011).

[40] F. Wu, E. Hwang, and S. Das Sarma, Phys. Rev. B 99,165112 (2019).

[41] I. Yudhistira, N. Chakraborty, G. Sharma, D. Y. H. Ho, E.Laksono, O. P. Sushkov, G. Vignale, and S. Adam, Phys.Rev. B 99, 140302(R) (2019).

[42] H. Ochoa, Phys. Rev. B 100, 155426 (2019).[43] T.-F. Chung, Y. Xu, and Y. P. Chen, Phys. Rev. B 98, 035425

(2018).[44] H. Polshyn, M. Yankowitz, S. Chen, Y. Zhang, K. Watanabe,

T. Taniguchi, C. R. Dean, and A. F. Young, arXiv:1902.00763.

[45] T. Sohier, M. Calandra, C.-H. Park, N. Bonini, N. Marzari,and F. Mauri, Phys. Rev. B 90, 125414 (2014).

[46] C.-H. Park, N. Bonini, T. Sohier, G. Samsonidze, B.Kozinsky, M. Calandra, F. Mauri, and N. Marzari, NanoLett. 14, 1113 (2014).

[47] P. T. Brown, D. Mitra, E. Guardado-Sanchez, R. Nourafkan,A. Reymbaut, C.-D. Hebert, S. Bergeron, A. M. S.Tremblay, J. Kokalj, D. A. Huse, P. Schauss, and W. S.Bakr, Science 363, 379 (2019).

[48] C. H. Mousatov, I. Esterlis, and S. A. Hartnoll, Phys. Rev.Lett. 122, 186601 (2019).

PHYSICAL REVIEW LETTERS 124, 076801 (2020)

076801-7